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Does this help? this is just an idea. You can improve it, of course.

var('x')
anti=integrate((4*x - sqrt(-x^2 + 1))/(sqrt(-x^2 + 1) + 5), x)
#anti=integrate(1/(sqrt(x + 1)*sqrt(-x + 1) + 5), x)
if str(anti).find("integrate") != -1:
    print("Result contains unevaluated integral!")
anti

Output:

Result contains unevaluated integral!
-x - 4*sqrt(-x^2 + 1) + 5*integrate(1/(sqrt(x + 1)*sqrt(-x + 1) + 5), x) + 20*log(sqrt(-x^2 + 1) + 5)